The two buckets have different temperature - so the gas inside the hose has different temperatures at the sides 1 and 2. But why isn't the volume in part 1 bigger than the volume in part 2? Is that because the pressure is not constant?

1) I have a ballon filled with helium at 30 degrees, and then I put it in the freezer. Then the volume changes, but the pressure stays constant, right?

Well why does the volume decrease?

More fundamentally, why is the balloon "inflated"? Because the air inside exerts a pressure that causes the fabric to expand and voila, inflated balloon. If you shove in too much air the pressure is too great and the balloon ruptures. Reduce the pressure and the balloon shrinks a bit. Normally you reduce the pressure by letting air out. Cooling it however will slow the molecules in the air, reducing their average kinetic energy, so they're not gonna spread out and cover as much space, and the pressure is reduced, which is why the volume decreases

For part B, it's as simple as the two buckets are the same size. You always assume the gas expands to fill its container, so there you have it.

EDIT: So you can infer everything you'd need to know from the ideal gas law. PV=nRT, even if it's not an ideal gas the basic relationships are the same

If you increase pressure while holding volume constant, temperature has to increase(so that the equality holds, you made the left side bigger, n and R are constants, gotta make the right side bigger) and similarly for all the relationships. Basically remember that in a gas the temperature is a measurement of average kinetic energy. If there's a high temperature the molecules are bouncing a lot harder and are gonna spread out and hit walls harder, meaning increased pressure, unless you allow the walls to expand, then increase volume. If you have gas with a set temperature and you shrink the volume, you have all those molecules with whatever kinetic energy now confined to a smaller space. If you're bouncing off the walls already, and you make the walls closer, you're gonna be bouncing harder, and so on